Solution of the 2-D steady-state radiative transfer equation in participating media with specular reflections using SUPG and DG finite elements

The 2D radiative transfer equation coupled with specular reflection boundary conditions is solved using finite element schemes. Both Discontinuous Galerkin and Streamline-Upwind Petrov–Galerkin variational formulations are fully developed. These two schemes are validated step-by-step for all involved operators (transport, scattering, reflection) using analytical formulations. Numerical comparisons of the two schemes, in terms of convergence rate, reveal that the quadratic SUPG scheme proves efficient for solving such problems. This comparison constitutes the main issue of the paper. Moreover, the solution process is accelerated using block SOR-type iterative methods, for which the determination of the optimal parameter is found in a very cheap way.